His Girl Friday is a reference to the Daniel Defoe novel Robinson Crusoe , written in 1719. One of the main characters in the book is a dude named Friday.
Crusoe, an Englishman, is stranded on an island; Friday is a native. Friday becomes Crusoe's servant (because Defoe was kind of a racist, and thought everyone should serve Englishman).
The phrase "my man Friday" came to mean "my servant," or "my guy who picks up after me—the subordinate who cleans my shoes and brushes my teeth and generally handles all the stuff that's too tedious for me to deal with."
The title of His Girl Friday switches the gender of the servant. This is a wink at the fact that Hildy's gender in the film is switched. The original version of His Girl Friday was a play from 1929 called The Front Page, in which Hildy is a man.
So His Girl Friday signals both that Hildy is a woman and that the role used to be for a man. It also, maybe, signals that Hildy is like a man in some ways —by being career-focused, for example, which was considered a masculine trait at the time.
The title also suggests that Walter and Hildy aren't equal. She's his servant or helper or right hand woman, not the other way around. He's not only her boss, he's the guy who wears the pants in the relationship.
And maybe His Girl Friday's also a joke about how stuck up Walter is; he does tend to treat everyone as if he's the lord of the island (or "the lord of the universe" as Hildy sarcastically calls him).
Or maybe the title is a more straightforward description of how Hawkes, and the film, sees Hildy—for all her success as a career woman, she's ultimately indebted to, and tied to, Walter.
The film certainly presents Hildy as independent and competent—but it also sees limits to that independence and competence. She is awesome, but she's still a girl Friday… emphasis on the "girl."